Non-existence of positive solutions to semilinear elliptic equations on \(\mathbb{R}^n\) or \(\mathbb{R}_+^n\) through the method of moving planes. (English) Zbl 0910.35048

The paper is mainly concerned with the semilinear elliptic problem \[ \Delta u+K \bigl(| x| \bigr)u^p=0 \quad \text{in } \mathbb{R}^n\;(n\geq 3), \quad u>0, \] where \(K(| x|)\) is nonnegative. The non-existence of a radially symmetric solution of (*) has been shown for \(p= (n+2)/(n-2)\), and for \(p>1\) under suitable monotonicity conditions on \(K(r)\). Here is proved the non-existence of a non-radial solution for the case where either \(p>(n+2)/(n-2)\) and \({d\over dr} (r^\sigma K(r)) \geq 0\), or \(1<p \leq (n+2)/(n-2)\), and \({d\over dr} (r^{\sigma/2} K(r))\geq 0\) and \(\not \equiv 0\), where \(\sigma =n+2- p(n-2)\). For the problem (*) with \(\mathbb{R}^n \setminus \{0\}\) instead of \(\mathbb{R}^n\), a \(C^2(\mathbb{R}^n \setminus \{0\})\) solution is proved to be radially symmetric under the monotonicity condition of \(K(r) | r^2- \overline r^2 |^{\sigma/2}\) in \((0,\overline r]\) and in \([\overline r,\infty)\) respectively.
The proofs are based on the Kelvin transformation and the repeated application of the moving plane method.


35J60 Nonlinear elliptic equations
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